A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<ol class="i-list"><li data-list-item-id="atan-30" data-list-item-path="atan-30"><div class="list-item-paragraph">Using a right-angled triangle, show that, for&nbsp;$a&gt;0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]<span class="inline-marks">[2]</span></div></li></ol>
<p>The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-33" data-list-item-path="atan-33"><div class="list-item-paragraph">Determine the values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=5$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-36" data-list-item-path="atan-36"><div class="list-item-paragraph">Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-37" data-list-item-path="atan-37"><div class="list-item-paragraph">Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part <strong>(i)</strong>.<span class="inline-marks">[5]</span></div></li></ol>

Using a right-angled triangle, show that, for $a>0$, \[\tan^{-1}(a)+\tan^{-1}\left(\frac1a\right)=\frac{\pi}{2}.\]

The curves $D_1$ and $D_2$ have equations \[D_1:\ y=\frac{2}{1+(x-1)^2},\qquad D_2:\ y=\frac{k}{12x-7},\] where $k>0$.

Determine the values of $k$ for which the curves have at least one point of intersection.

Sketch both curves on the same axes. Label their intersections with the coordinate axes, the maximum point of $D_1$, and all asymptotes.

Calculate the exact area of the finite region enclosed by the curves, simplifying your answer using part **(i)**.

Sketching Rational Functions

Inverse Trigonometric Functions

Parameter Conditions for Graphs and Equations

Asymptotes and Discontinuities

Area bounded by 2 curves

A complementary-angle identity and a shifted-curve area

Applications of Integration

Graphing Techniques

Functions

<ol class="i-list"><li data-list-item-id="atan-21" data-list-item-path="atan-21"><div class="list-item-paragraph">By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]<span class="inline-marks">[1]</span></div></li></ol>
<p>The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where&nbsp;$k&gt;0$.</p>
<ol class="i-list" start="2"><li data-list-item-id="atan-24" data-list-item-path="atan-24"><div class="list-item-paragraph">Find the set of values of $k$ for which the curves have at least one point of intersection.<span class="inline-marks">[3]</span></div></li></ol>
<p>For the remaining parts, take $k=\frac32$.</p>
<ol class="i-list" start="3"><li data-list-item-id="atan-27" data-list-item-path="atan-27"><div class="list-item-paragraph">Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="atan-28" data-list-item-path="atan-28"><div class="list-item-paragraph">Find the exact area of the finite region enclosed by the curves. Use the identity in part <strong>(i)</strong> to simplify your answer.<span class="inline-marks">[5]</span></div></li></ol>

By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\]

The curves $C_1$ and $C_2$ have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where $k>0$.

Find the set of values of $k$ for which the curves have at least one point of intersection.

For the remaining parts, take $k=\frac32$.

Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.

Find the exact area of the finite region enclosed by the curves. Use the identity in part **(i)** to simplify your answer.

An inverse-tangent identity and an enclosed rational-curve area

<p>A curve $C$ has equation $y=\dfrac{x^2-4x+1}{x+1}$, $x \ne -1$.</p>
<ol class="abc-list"><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-0" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-0"><div class="list-item-paragraph">Prove, using an algebraic method, that&nbsp;$y=\dfrac{x^2-4x+1}{x+1}$ cannot lie between two certain values, to be determined in exact form.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-1" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-1"><div class="list-item-paragraph">Sketch $C$, stating the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2"><div class="list-item-paragraph">The curve $C$ undergoes, in succession, the following transformations.</div><ol class="num-list"><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-0" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2/h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-0"><div class="list-item-paragraph">A translation of 1 unit in the positive $x$-direction,</div></li><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-1" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2/h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-1"><div class="list-item-paragraph"><p>A stretch parallel to the $x$-axis, factor $3$, $y$-axis invariant,</p></div></li><li data-list-item-id="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-2" data-list-item-path="h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2/h2-2025-vjc-promo-q9-vjc-q9-item-block-1-item-2-sub-2"><div class="list-item-paragraph">A reflection in the $y$-axis.</div></li></ol><div class="list-item-after-content">Find the equation of the new curve in the form $y=\mathrm{f}(x)$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y=\dfrac{x^2-4x+1}{x+1}$, $x \ne -1$.

Prove, using an algebraic method, that $y=\dfrac{x^2-4x+1}{x+1}$ cannot lie between two certain values, to be determined in exact form.

Sketch $C$, stating the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.

Composite Transformations

Finding Possible y-values of a Curve

2025 VJC Promo Q9

Transformations of Graphs

<p>A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where&nbsp;$x\ne -4$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the $x$- and $y$-intercepts of the graph of $y=f(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=f(x)$, clearly indicating the asymptotes and intercepts.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Hence, sketch the graph of $y=f(|x|)$ on a separate set of axes, clearly indicating any asymptotes and axial intercepts.<span class="inline-marks">[3]</span></div></li></ol>

A function is defined by $f(x)=\dfrac{3x-2}{x+4}$, where $x\ne -4$.

Sketching Graphs of $f(\lvert x\rvert)$

Transformations of a rational function and modulus

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Sketch the curve with equation $y=\left|\frac{4}{p-x}\right|$, where&nbsp;$p$ is a positive constant. State, in terms of $p$, the equations of the asymptotes and the coordinates of any intersections with the coordinate axes. On the same diagram, sketch the line with equation $y=q(x-p)$, where&nbsp;$q$ is a positive constant.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find, in terms of $p$ and $q$, the root of the equation $\left|\frac{4}{p-x}\right|=q(x-p)$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Hence solve the inequality $\left|\frac{4}{p-x}\right|&gt;q(x-p)$.<span class="inline-marks">[2]</span></div></li></ol>

Graphical Method

Inequalities involving Rational Functions

Graphs of Modulus Function

Inequalities involving Modulus Functions

Modulus Rational Function and Linear Intersection

Inequalities

<p>A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Sketch $C$, indicating on the diagram the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.<span class="inline-marks">[4]</span></div></li></ol>
<p>A curve $D$ has parametric equations</p>
<p style="text-align: center">\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for}\hspace{0.5em} -\frac{\pi}{2} &lt; \theta &lt; \frac{\pi}{2}.\]</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the cartesian equation of curve $D$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Hence, on the same diagram as in part <strong>(a)</strong>, sketch the curve $D$ and determine the number of intersection points between curve $C$ and curve $D$.<span class="inline-marks">[3]</span></div></li></ol>

A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$.

\[x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for } -\frac{\pi}{2} < \theta < \frac{\pi}{2}.\]

Graphical Root Counting

Sketching Parametric Curves

Converting Parametric Equations to Cartesian Form

2024 EJC P1 Q4

<p>The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where&nbsp;$a$ is a constant, $a \neq 6$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Find the set of values of $a$ such that&nbsp;$C$ does not cut the $x$-axis.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">It is given that the equation of $C$ is $y = \frac{3x^2 - 6x + 3}{x + 1}$. Sketch $C$ and give the equations of any asymptotes. Also, state the coordinates of any points where&nbsp;$C$ crosses the axes and of any turning points.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c"><div class="list-item-paragraph">Let&nbsp;$k$ be a positive constant. By sketching a suitable graph in the same diagram in part <strong>(b)</strong>, find an inequality satisfied by $k$ such that the equation</div><div class="list-item-paragraph" style="text-align: center">\[k^2(x - 1)^2 + \left(\frac{3x^2 - 6x + 3}{x + 1} + 12\right)^2 = k^2\]</div><div class="list-item-paragraph">has 2 real and distinct roots.<span class="inline-marks">[2]</span></div></li></ol>

The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$, where $a$ is a constant, $a \neq 6$.

2023 VJC Promo Q10

<p>The curve $C_1$ has equation $y = 2x + 3 + \frac{12}{x - 2}$, $x \neq 2$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Sketch $C_1$, showing clearly the equations of asymptotes, the coordinates of the axial intercepts and the turning points, if any.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">By completing the square for both&nbsp;$x$ and $y$, give a geometrical interpretation of the curve $C_2$ with equation $x^2 - 4x + y^2 + 6y - 3 = 0$.</div><div class="list-item-paragraph">Hence write down the number of real roots for the equation \[(x - 2)^2 + \left(2x + 6 + \frac{12}{x - 2}\right)^2 = 4^2.\]<span class="inline-marks">[3]</span></div></li></ol>

The curve $C_1$ has equation $y = 2x + 3 + \frac{12}{x - 2}$, $x \neq 2$.

Graphs of Conic Sections

2024 TMJC Promo Q3

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \mapsto \frac{1}{(x-2)^2+1},\quad x \in \mathbb{R}.\]</p>
<ol class="i-list"><li><div class="list-item-paragraph">Sketch the graph of $y = \mathrm{f}(x)$, indicating the coordinates of the turning point and the equation of any asymptotes. Explain why $\mathrm{f}^{-1}$ does not exist.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">If the domain of $\mathrm{f}$ is restricted to $[k, \infty)$, state the smallest value of $k$ for which the function $\mathrm{f}^{-1}$ exists.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Using the value of $k$ found in part <strong>(ii)</strong>, find $\mathrm{f}^{-1}(x)$ and state its domain.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}$ is defined by

\[\mathrm{f} : x \mapsto \frac{1}{(x-2)^2+1},\quad x \in \mathbb{R}.\]

Restricting a Domain to Obtain an Inverse

Inverse Functions

Horizontal Line Test

2024 MI PRU2 P2 Q2

<p>The curve $C$ has equation</p>
<p style="text-align: center">\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where}\hspace{0.5em} k &gt; 0, \ k \ne 2.\]</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Without the use of a calculator, find the coordinates of any points of intersection of $C$ with the axes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Write down the equations of the asymptotes of $C$ in terms of $k$.<span class="inline-marks">[2]</span></div></li></ol>
<p>It is now given that&nbsp;$k = 1$.</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Show that the $x$-coordinates of the turning points of $C$ satisfy the equation \[x^4 - 7x^2 + 12x - 2 = 0.\] Hence find these $x$-coordinates.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Sketch $C$, labelling all essential features.<span class="inline-marks">[3]</span></div></li></ol>

\[y = x - 2 + \frac{3k(x - 2)}{x^2 - 2k}, \quad \text{where } k > 0, \ k \ne 2.\]

Quotient Rule

Stationary Points and Their Nature

2024 EJC Promo Q12 

Applications of Differentiation

Techniques of Differentiation

<p>The function $\mathrm{f}$ is defined by</p>
<p style="text-align: center">\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence state the range of $\mathrm{f}$.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Explain why the function $\mathrm{f}^2$ does not exist.</span><span class="inline-marks">[1]</span></div></li></ol>
<p>The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where&nbsp;$a$ is a constant.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find $\mathrm{f}^{-1}(x)$ and state its domain.</span><span class="inline-marks">[4]</span></div></li></ol>

\[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.\]

Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.

Explain why the function $\mathrm{f}^2$ does not exist.

The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where $a$ is a constant.

State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.

Hence find $\mathrm{f}^{-1}(x)$ and state its domain.

Existence of Composite Functions

Modulus Functions

Range of a Function

N2023 P1 Q7

<p>The curve $C_1$ is given by the equation</p>
<p style="text-align: center">\[y = \frac{ax^2 + bx}{x + c}\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">The curve $C_1$ has a vertical asymptote at $x = 2$. The linear oblique asymptote of $C_1$ passes through the points with coordinates $(0, 10)$ and $(-2, 0)$. Determine the values of the unknown constants $a$, $b$ and $c$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence sketch $C_1$, stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).</span></div></li></ol>

The curve $C_1$ is given by the equation

The curve $C_1$ has a vertical asymptote at $x = 2$. The linear oblique asymptote of $C_1$ passes through the points with coordinates $(0, 10)$ and $(-2, 0)$. Determine the values of the unknown constants $a$, $b$ and $c$.

Hence sketch $C_1$, stating clearly the equation(s) of any asymptote(s) and the coordinates of any stationary point(s).

2014 NJC MYE Q7

<p>Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$y = 2x^3 - 5x^2 + x - 2$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = 4x^2 e^x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{\ln x}{2x}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = |2x - x^2|$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$-1 &lt; a &lt; 0$ and $0 &lt; b &lt; 1$;</span></div></li><li><div class="list-item-content"><span class="list-item-text">$a &lt; -1$ and $b &gt; 1$.</span></div></li></ol></li></ol>
<p>What could be said about the graph if&nbsp;$ab + 1 = 0$?</p>

Sketch the graph represented by each of the following equations, clearly indicating the equations of asymptote(s), axial intercept(s), and the coordinates of the turning point(s), if any.

$y = \frac{x^2 + 2x - 8}{x^2 + 2x - 3}$

$y = \frac{ax + 1}{x - b}$, where constants $a$, $b$ are given by

What could be said about the graph if $ab + 1 = 0$?

Sketching Exponential Functions and Solving Equations Graphically

Sketching Logarithmic Functions and Solving Equations Graphically

Polynomial Functions and Key Features

<p>The curve $D$ has equation $y = \frac{2x - 5}{x^2 + 2x - 3}.$</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of $D$, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">By considering a suitable curve, hence, find the exact range of values of $k$, where&nbsp;$k &gt; 0$, such that the equation $(x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2$ has at least one <strong>positive</strong> real root.<span class="inline-marks">[3]</span></div></li></ol>

The curve $D$ has equation $y = \frac{2x - 5}{x^2 + 2x - 3}.$

Sketch the graph of $D$, stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s).

By considering a suitable curve, hence, find the exact range of values of $k$, where $k > 0$, such that the equation \[(x + 3)^2 + \left( \frac{2x - 5}{x^2 + 2x - 3} \right)^2 = k^2\] has at least one **positive** real root.

Circle–Curve Intersection Conditions

2024 NYJC Promo Q3

<p>The curve $C$ has equation $y = \frac{x^2 + 3x}{x - 1}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the exact set of values of $y$ for which there are no point on $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Draw a sketch of $C$, showing clearly any axis intercepts, asymptotes and turning points.</span></div></li><li><div class="list-item-content"><span class="list-item-text">In the same diagram, draw a sketch of the curve $y = (x - 5)^2 + 1$ and hence find the number of real roots of the equation $x^3 - 12x^2 + 33x - 26 = 0$.</span></div></li></ol>

The curve $C$ has equation $y = \frac{x^2 + 3x}{x - 1}$.

Find the exact set of values of $y$ for which there are no point on $C$.

Draw a sketch of $C$, showing clearly any axis intercepts, asymptotes and turning points.

In the same diagram, draw a sketch of the curve $y = (x - 5)^2 + 1$ and hence find the number of real roots of the equation $x^3 - 12x^2 + 33x - 26 = 0$.

N1990 P1 Q1

<p>The curve $C$ is given by the equation $y=\frac{1}{x}+\frac{2}{x^2}, x \neq 0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, find the set of values that&nbsp;$y$ can take.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the curve $C$, stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>Given that the solution of the inequality $ax^2+bx+c &gt; \frac{1}{x}+\frac{2}{x^2}$ is the set&nbsp;$\{x \in \mathbb{R} : -1.5 &lt; x &lt; -1 \text{ or}\hspace{0.5em} 1 &lt; x &lt; k\}$, find the values of $a$, $b$ and $c$.<span class="inline-marks">[3]</span></p>
<p>Hence find the value of $k$.<span class="inline-marks">[1]</span></p>

The curve $C$ is given by the equation $y=\frac{1}{x}+\frac{2}{x^2}, x \neq 0$.

Without using a calculator, find the set of values that $y$ can take.

Sketch the curve $C$, stating the equations of any asymptotes and the coordinates of any turning points and points of intersection with the axes.

Given that the solution of the inequality $ax^2+bx+c > \frac{1}{x}+\frac{2}{x^2}$ is the set $\{x \in \mathbb{R} : -1.5 < x < -1 \text{ or } 1 < x < k\}$, find the values of $a$, $b$ and $c$.

Determining Model Coefficients Using Linear Systems

2015 ACJC P1 Q9

<p>The equation of a curve $C$ is given by $y=1+\frac{6}{x-3}-\frac{24}{x+3}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the equations of the asymptotes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of the points where&nbsp;$C$ meets the axes.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, find the set of values of $y$ for which there are no points on $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch $C$.</span></div></li></ol>

The equation of a curve $C$ is given by $y=1+\frac{6}{x-3}-\frac{24}{x+3}$.

Write down the equations of the asymptotes.

Find the coordinates of the points where $C$ meets the axes.

Without using a calculator, find the set of values of $y$ for which there are no points on $C$.

<p>Write down all the asymptote(s) and axial intercepts of $y=\frac{5{{x}^{2}}+1}{{{x}^{2}}-3x}$ and sketch the curve.</p>

Write down all the asymptote(s) and axial intercepts of $y=\frac{5{{x}^{2}}+1}{{{x}^{2}}-3x}$ and sketch the curve.

<p>A curve $C$ has equation $y = \frac{a}{2bx - x^2}$, where&nbsp;$a, b &gt; 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Describe the transformation that maps the graph of $C$ onto the graph of $y = \frac{a}{b^2 - x^2}$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$a = 1$ and $b = 2$, sketch $C$ stating the equations of any asymptotes and coordinates of any stationary points and of the points where the curve crosses the axes.</span><span class="inline-marks">[2]</span></div></li></ol>

A curve $C$ has equation $y = \frac{a}{2bx - x^2}$, where $a, b > 0$.

Describe the transformation that maps the graph of $C$ onto the graph of $y = \frac{a}{b^2 - x^2}$.

Given that $a = 1$ and $b = 2$, sketch $C$ stating the equations of any asymptotes and coordinates of any stationary points and of the points where the curve crosses the axes.

Translations, Reflections and Scalings of Graphs

2025 TJC P1 Q3

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">On a single diagram, sketch the graphs of $y = \frac{3x}{x - 1}$ and $y = 3x$, stating the equation(s) of any asymptote(s) and the coordinates of points of intersection.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">Hence find the area bounded by the curve $y = \frac{3x}{x - 1}$ and the lines $y = 3x$ and $y = 8$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">A curve has the equation $y = \sqrt{x^2 + 1}$ where&nbsp;$x$ is non-negative.</div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-paragraph">With the aid of a sketch of the above curve, explain why the value of $S = \lim_{n \to \infty} \frac{1}{n} \left[ 1 + \sqrt{\frac{1}{n^2} + 1} + \sqrt{\frac{4}{n^2} + 1} + \dots + \sqrt{\frac{(n - 1)^2}{n^2} + 1} \right]$ is $\displaystyle \displaystyle\int_0^1 \sqrt{x^2 + 1} \, \mathrm{d}x$.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-paragraph">Hence find the value of $S$.<span class="inline-marks">[1]</span></div></li></ol></li></ol>

A curve has the equation $y = \sqrt{x^2 + 1}$ where $x$ is non-negative.

Math Question Bank